Covering homotopy

for a homotopy of a mapping , given a mapping

A homotopy such that . In this situation, if the covering mapping for is prescribed in advance, one says that extends . The covering homotopy axiom, in its strong version, requires that, for a given mapping , for any homotopy from a paracompactum and for any (), an extension of to a covering homotopy exists. In that case is said to be a Hurewicz fibration. The most important example is provided by the locally trivial fibre bundles (cf. Locally trivial fibre bundle). If the covering homotopy property is only required to hold in the case that is a finite polyhedron, is called a Serre fibration.

Let and be arcwise connected and let be the path space of (i.e. the space of continuous mappings ). Consider a continuous mapping

where

and assume that begins at a point and covers . Then the formula yields an extension of to a covering homotopy . In particular, a mapping satisfying these conditions can be defined naturally for a covering, and also for a smooth vector bundle with a fixed connection. The validity of the covering homotopy axiom in Serre's formulation makes it possible to construct the exact homotopy sequence of a fibration (see Homotopy group).

Comments

Thus, a covering homotopy is a lifting of a given homotopy (a homotopy lifting). The covering homotopy property is dual to the homotopy extension property, which defines the notion of a cofibration.

References

[a1]	E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) pp. Chapt. 2